{smcl}
{* 3oct2012}{...}
{hline}
help for {hi:lmoments}
{hline}
{title:L-moments and derived statistics}
{p 8 17 2}{cmd:lmoments}
[{it:varlist}]
[{cmd:if} {it:exp}]
[{cmd:in} {it:range}]
[{cmd:,}
{cmdab:all:obs}
{cmd:lmax(}{it:#}{cmd:)}
{cmd:short}
{help tabdisp:tabdisp_options}
{cmd:variablenames}
{cmd:saving(}{it:filename}[{cmd:,} {help save:save_options}{cmd:)}
]
{p 8 17 2}{cmd:lmoments}
{it:varname}
[{cmd:if} {it:exp}]
[{cmd:in} {it:range}]
[{cmd:,}
{cmd:by(}{it:byvarlist}{cmd:)}
{cmdab:miss:ing}
{cmd:lmax(}{it:#}{cmd:)}
{cmd:short}
{help tabdisp:tabdisp_options}
{cmd:saving(}{it:filename}[{cmd:,} {help save:save_options}{cmd:)}
]
{p 4 4 2}{cmd:by ... :} may also be used with {cmd:lmoments}:
see help on {help by}.
{title:Description}
{p 4 4 2}{cmd:lmoments} calculates L-moments and derived statistics for
a {it:varlist}. Any string variables in {it:varlist} are ignored.
Specifically, and by default, the first four L-moments and the derived
statistics t, t_3 and t_4 are calculated for each variable in
{it:varlist}.
{title:Remarks}
{p 4 4 2}Denote by X(j:n) the j th smallest observation from a sample of
size n from a variable X and by E the expectation operator.
{p 4 4 2}The first four L-moments are defined by
E (X(1:1)),
1/2 E (X(2:2) - X(1:2)),
1/3 E (X(3:3) - 2 X(2:3) + X(1:3)) and
1/4 E (X(4:4) - 3 X(3:4) + 3 X(2:4) - X(1:4)).
{p 4 4 2}They are estimated via these weighted averages for a sample
x_1, ..., x_n, otherwise known as probability-weighted moments:
b_0 = average of x(j:n),
j - 1
b_1 = average of {hline 5} x(j:n),
n - 1
j - 1 j - 2
b_2 = average of {hline 5} {hline 5} x(j:n) and
n - 1 n - 2
j - 1 j - 2 j - 3
b_3 = average of {hline 5} {hline 5} {hline 5} x(j:n).
n - 1 n - 2 n - 3
{p 4 4 2}
The estimators are
l_1 = b_0,
l_2 = 2 b_1 - b_0,
l_3 = 6 b_2 - 6 b_1 + b_0 and
l_4 = 20 b_3 - 30 b_2 + 12 b_1 - b_0,
{p 4 4 2}whence
t = l_2 / l_1 (cf. coefficient of variation),
t_3 = l_3 / l_2 (cf. skewness) and
t_4 = l_4 / l_2 (cf. kurtosis).
{title:Options}
{p 4 8 2}{cmd:allobs} specifies use of the maximum possible number of
observations for each variable. The default is to use only those
observations for which all variables in {it:varlist} are not missing.
{p 4 8 2}{cmd:by()} specifies one or more variables defining distinct
groups for which L-moments should be calculated. {cmd:by()} is allowed
only with a single {it:varname}. The choice between {cmd:by:} and
{cmd:by()} is partly one of precisely what kind of output display is
required. The display with {cmd:by:} is clearly structured by groups
while that with {cmd:by()} is more compact. To show L-moments for
several variables and several groups with a single call to
{cmd:lmoments}, the display with {cmd:by:} is essential.
{p 4 8 2}{cmd:missing} specifies that, if {cmd:by()} is specified,
observations with missing values on {it:byvarlist} are to be included in
calculations. The default is to exclude them. Missing values on
{it:varlist} are always and necessarily ignored.
{p 4 8 2}{cmd:lmax()} specifies calculation of the measures l_5 upwards
to the specified maximum and correspondingly of the measures t_5 upwards
in addition to the default. Thus {cmd:lmax(8)} adds L-moments 5, 6, 7
and 8 and ratios t_5, ..., t_8. See the references for definitions.
Results are not displayed, but may be saved to a new dataset via the
{cmd:saving()} option. This is a rarely specified option for those
exploring the uses of these measures.
{p 4 8 2}{cmd:short} specifies display of n, l_1, l_2, t_3, t_4 only.
This option has no effect on the calculation.
{p 4 8 2}{it:tabdisp_options} are options of {help tabdisp}. The
default display has {cmd:format(%9.3f)}.
{p 4 8 2}{cmd:variablenames} specifies that the variable names of
{it:varlist} should be used in display. The default is to use variable
labels to indicate a set of variables.
{p 4 8 2}{cmd:saving()} specifies a filename in which to save the
results of calculations as a Stata dataset. Optionally, the options of
{help save} itself may be specified.
{title:Examples}
{p 4 8 2}{cmd:. sysuse auto, clear}
{p 4 8 2}{cmd:. lmoments, short}
{p 4 8 2}{cmd:. lmoments price-foreign}
{p 4 8 2}{cmd:. bysort rep78: lmoments mpg}
{p 4 8 2}{cmd:. lmoments mpg, by(rep78) missing}
{p 4 8 2}{cmd:. lmoments mpg, by(rep78) missing saving(lmoresults, replace)}
{title:Saved results}
{p 4 4 2}(all for last-named variable or group only)
r(N) n
r(l_1) l_1
r(l_2) l_2
r(l_3) l_3
r(l_4) l_4
... (higher sample L-moments if requested)
r(t) t
r(t_3) t_3
r(t_4) t_4
... (higher sample L-moment ratios if requested)
{title:Acknowledgments}
{p 4 4 2}
{cmd:lmoments} is a descendant of Patrick Royston's {cmd:lshape} program.
Stephen Jenkins found a bug in a previous version of this program.
{title:Author}
{p 4 4 2}Nicholas J. Cox, Durham University, U.K.{break}
n.j.cox@durham.ac.uk
{title:References}
{p 4 8 2}{browse "http://researcher.watson.ibm.com/researcher/view_project.php?id=1021":L-moments}
{p 4 8 2}Hosking, J.R.M. 1990. L-moments: Analysis and estimation of
distributions using linear combinations of order statistics.
{it:Journal of the Royal Statistical Society} Series B 52: 105{c -}124.
{p 4 8 2}Hosking, J.R.M. 1998. L-moments. In Kotz, S., C.B. Read and
D.L. Banks (eds) {it:Encyclopedia of Statistical Sciences Update Volume 2.}
New York: Wiley, 357{c -}362.
{p 4 8 2}Hosking, J.R.M. 2006.
On the characterization of distributions by their L-moments.
{it:Journal of Statistical Planning and Inference}
136: 193{c -}198.
{p 4 8 2}Hosking, J.R.M. and J.R. Wallis. 1997.
{it:Regional frequency analysis: an approach based on L-moments.}
Cambridge University Press.
{p 4 8 2}Jones, M.C. 2004.
On some expressions for variance, covariance, skewness and L-moments.
{it:Journal of Statistical Planning and Inference}
126: 97{c -}106.
{p 4 8 2}Royston, P. 1992.
Which measures of skewness and kurtosis are best?
{it:Statistics in Medicine} 11: 333{c -}343.
{p 4 8 2}Serfling, R. and Xiao, P. 2007.
A contribution to multivariate L-moments: L-comoment matrices.
{it:Journal of Multivariate Analysis} 98: 1765{c -}1781.
{title:See also}
{p 4 8 2}{help lmose} (if installed)